Method and decoder for reconstructing a source signal

ABSTRACT

In a method for reconstructing a source signal, which is encoded by a set of at least two descriptions, the method comprises: receiving a subset of the set of descriptions; reconstructing a reconstructed signal at an operating bitrate of a set of operating bitrates upon the basis of the subset of descriptions, the reconstructed signal having a second probability density, wherein the second probability density comprises a first statistical moment and a second statistical moment; and manipulating the reconstructed signal, wherein the reconstructed signal is manipulated such that, irrespective of the operating bitrate, a predetermined minimum similarity between a first statistical moment of a third probability density and a first statistical moment of a first probability density and between a second statistical moment of the third probability density and a second statistical moment of the first probability density is maintained.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/087,080, filed on Nov. 22, 2013, which is a continuation ofInternational Application No. PCT/CN2011/074508, filed on May 23, 2011.The afore-mentioned patent applications are hereby incorporated byreference in their entireties.

FIELD

The invention relates to the field of digital signal processing and inparticular to multiple description coding (MDC).

BACKGROUND

For modern data communications, a plurality of network technologiesenabling wireless networks or fixed line networks may be deployed. Suchnetworks may in some cases not be reliable, such that data orinformation, respectively, is lost during transmission. In particularfor multimedia content, a graceful quality degradation may be acceptablein the event of such information loss during transmission. To this end,MDC aims at achieving an acceptable quality degradation by generating aplurality of mutually refinable descriptions of a single signal source.These descriptions are transmitted from an encoder to a decoder,typically independently of each other. The descriptions have theproperty that generally any subset of the descriptions may make areconstruction of the source signal possible. In particular, the moredescriptions are available to the decoder, the better the reconstructioncan be. As a consequence, MDC facilitates multiple quality levelsdepending on a state of a transmission channel of an unreliablecommunication network.

In conventional MDC decoders, the source signal may be reconstructedbased on any possible subset of descriptions. The estimation of thesource signal may be based on a minimization of a mean-squared error(MSE). However, the quality of reconstruction depends on a bitrate, atwhich the descriptions are transmitted, wherein a reconstruction qualityincreases with increasing bitrate. The usage of MSE may be perceptuallymotivated only at low distortions or equivalently at high bitrates.Hence, for lower bitrates, an obtained perceptual quality of thereconstructed signal using an MSE may be insufficient.

Furthermore, statistical properties of the signal source may be lost oraltered after reconstruction with a mean-squared error-based scheme.This may further negatively affect the quality of the reconstructedsignal.

SUMMARY

It is the object of the present invention to provide an improved conceptfor reconstructing a source signal that is encoded by multipledescriptions.

This object is achieved by the features of the independent claims.Further implementation forms are apparent from the dependent claims, thedescription and the drawings.

The invention is based on the finding that a source signal may beencoded with a set of descriptions, which are transmitted, for exampleover a communication network. The descriptions may be generated with anMDC encoder encoding a source signal which comprises a first probabilitydensity, which is fully or partially known or estimated at a decoderside. A subset of the set of descriptions can, at a decoder, be used toreconstruct a reconstructed signal comprising a second probabilitydensity. By manipulating the reconstructed signal, a manipulatedreconstructed signal is obtained, which comprises a third probabilitydensity. Through statistically manipulating the reconstructed signal,the third probability density can be amended such that the thirdprobability density is equal or similar to the first probability densityof the source signal. The manipulation is chosen such that, irrespectiveof an operating bitrate during the reconstruction, a minimum similaritybetween the third probability density and the first probability densityis maintained or achieved, respectively. Hence, the statisticalproperties of the manipulated reconstructed signal, in particular thesimilarity of the third probability density to the first probabilitydensity, may result in an improved quality of the manipulatedreconstructed signal for all operating bitrates, and in particular alsofor low operating bitrates.

According to a first aspect, the invention relates to a method forreconstructing a source signal which is encoded by a set of at least twodescriptions, the source signal having a first probability density,wherein the first probability density comprises a first statisticalmoment and a second statistical moment, the method comprising receivinga subset of the set of descriptions; reconstructing a reconstructedsignal at an operating bitrate of a set of operating bitrates upon thebasis of the subset of descriptions, the reconstructed signal having asecond probability density, wherein the second probability densitycomprises a first statistical moment and a second statistical moment;and manipulating the reconstructed signal in order to obtain amanipulated reconstructed signal having a third probability density,wherein the third probability density comprises a first statisticalmoment and a second statistical moment, wherein the reconstructed signalis manipulated such that at least the first statistical moment and thesecond statistical moment of the third probability density are moresimilar to the first statistical moment and the second statisticalmoment of the source signal than the first statistical moment and thesecond statistical moment of the second probability density are, andwherein furthermore the reconstructed signal is manipulated such that,irrespective of the operating bitrate, a predetermined minimumsimilarity between the first statistical moment of the third probabilitydensity and the first statistical moment of the first probabilitydensity and between the second statistical moment of the thirdprobability density and the second statistical moment of the firstprobability density is maintained.

The operating bitrate may be the total bitrate that is distributed amongthe descriptions, for example the bitrate available to a quantizer usedin an encoding or decoding process. The bitrate may be distributedarbitrarily between the descriptions. According to some implementationforms of the first aspect, the allocation of the bitrate among thedescriptions is symmetrical, such that all the descriptions areallocated with the same bitrate. However, each description may beassociated with a different bitrate, so that the total available bitrateis distributed among the descriptions.

The term “quantizer” denotes a mapping from a signal value, e.g. asample, to be quantized, which may be taken presumably from a continuousset, to a quantization index which is taken from a discrete set ofindices. Correspondingly, the term “dequantizer” denotes a mapping froma quantization index taken from a discrete set of indices or from asubset of indices taken from the discrete set of indices to areconstruction value.

The first statistical moment of the various probability densities is themean value associated with the probability density. The secondstatistical moment of the various probability densities may be thevariance associated with the probability density. The similarity betweenthe statistical moments may be defined as an absolute difference betweenthe respective statistical moments, for example between the means of theprobability densities or between the variances of the probabilitydensities. According to some implementation forms of the first aspect,the similarity may be defined as a normalized absolute difference, whichmay be the absolute difference of the statistical moments normalizedwith respect to the respective moment of the first probability density.

Regarding the predetermined minimum similarity, such value may bedefined as a percentage value or a decibel (dB) value. According to someimplementation forms, a deviation of 50% or 3 dB, respectively, may beacceptable and used as the predetermined minimum similarity.Furthermore, the predetermined minimum similarity may be different forthe first statistical moment and the second statistical moment. Inparticular, a similarity for the first statistical moment may beregarded as having a higher importance than the similarity for thesecond statistical moment. Therefore, a smaller difference value may bedesirable for the first statistical moment than for the secondstatistical moment. Accordingly, even a difference equal to zero can beset for the predetermined minimum similarity for the first statisticalmoment.

According to some implementation forms of the first aspect, thereconstructing of the source signal for all operating bitrates isimproved when compared to a conventional MDC decoder that is based on amean-squared error scheme. Thus, the described implementation formsrender an optimal MSE performance subject to a constraint that thedistribution of the reconstructed signal that is similar to the sourcedistribution. This leads to improving a perceptual quality of thereconstruction. The improved quality is maintained also for loweroperating bitrates, which is not possible in conventional MDC decodingmethods that use solely the mean-squared error.

According to a first implementation form of the first aspect, theinvention relates to a method for reconstructing a source signal,wherein the first statistical moment and the second statistical momentof the second probability density are manipulated to preserve the firststatistical moment and the second statistical moment of the firstprobability density within a predetermined moment range. Accordingly,the manipulating is performed such that the first and the secondstatistical moment of the first probability density are at leastpartially preserved in the manipulated reconstructed signal within agiven, predetermined range. According to some implementation forms, themanipulating is performed by means of an approximated transformationfunction, which can be implemented with less computational effort.However, according to some implementation forms, the moment range can bechosen such that the statistical moments of the first probabilitydensity are achieved within the third probability density of themanipulated reconstructed signal.

According to a second implementation form of the first aspect, theinvention relates to a method for reconstructing a source signal,wherein the first statistical moments of the first probability densityand the third probability density are equal, and wherein the secondstatistical moments of the first probability density and the thirdprobability density are equal.

According to a third implementation form of the first aspect, theinvention relates to a method for reconstructing a source signal,wherein the reconstructed signal is reconstructed using a reconstructionfunction that is dependent on a composition of descriptions in thesubset of descriptions. Accordingly, if all descriptions of the set ofdescriptions are received, such that the subset of descriptions is equalto the (full) set of descriptions, reconstruction can be performed onthe basis of the full set of descriptions. For example, the descriptionsare indices or pointers, which unambiguously lead to a reconstructionvalue to be used in the reconstruction signal. However, if at least oneof the descriptions of the set of descriptions is not received, adifferent reconstruction function will be used. In particular, thedescriptions are mutually refinable.

According to a fourth implementation form of the first aspect, theinvention relates to a method for reconstructing a source signal,wherein the source signal comprises an additive dither signal, inparticular a pseudo-random dither signal, and wherein the reconstructingcomprises subtracting the dither signal from the reconstructed signal.Such dither signals may have been added to the source signal duringencoding and may facilitate the derivation of distribution preservingtransformations. After reconstructing the reconstructed signal, thedither signal can then be subtracted in order to achieve thereconstructed signal. Adding and subtracting operations may be exchangedaccording to various implementation forms.

According to a fifth implementation form, the invention relates to amethod for reconstructing a source signal, wherein the reconstructingcomprises using an index assignment scheme, which is addressed by thedescriptions of the set of descriptions, the index assignment schemebeing used for deriving the set of descriptions encoding the sourcesignal. For example, the index assignment scheme may be represented bymeans of an index assignment matrix that is two- or more-dimensional,wherein the index assignment matrix consists of non-empty entriescontaining unique, so-called central indices and empty entries. Duringencoding, indices pointing to such a unique value are determined andchosen to be the descriptions for the encoded value of the sourcesignal. Accordingly, during reconstruction, the received descriptionsmay point to a value, if all descriptions are received, or a set ofvalues, if one or more descriptions are lost, within the indexassignment scheme or matrix, respectively. If one or more descriptionsare lost, a reconstruction value associated with the set of values maybe chosen, for example, a most probable value for the set of values.

According to a sixth implementation form of the first aspect, theinvention relates to a method for reconstructing a source signal,wherein the manipulating the reconstructed signal comprises transformingthe reconstructed signal according to a statistical transformationfunction, the transformation function being dependent on a compositionof descriptions in the subset of descriptions. Hence, the transformationfunction is determined from the knowledge, which descriptions of the setof descriptions are received and which descriptions may be lost.

According to some implementation forms, the transformation function T(x)is defined according to the following formula:

${{T(x)} = {F_{X}^{- 1}\{ {\frac{1}{\Delta}{\int_{- \frac{\Delta}{2}}^{\frac{\Delta}{2}}{{F_{X}( {x - \tau} )}\ {\mathbb{d}\tau}}}} \}}},$where Δ is a quantizer step size, F_(X) (x) is the cumulativedistribution function of variable X that is related to the probabilitydensity function ƒ_(X)(·) of the first probability density, as

F_(X)(x) = ∫_(−∞)^(x)f_(X)(τ) 𝕕τ,and F_(X) ⁻¹(·) denotes the inverse cumulative distribution function.

The probability density function may be known in the decoding process inadvance. The above transformation function T(x) may be used if all, forexample two, descriptions are received, and the first probabilitydistribution should be preserved exactly or almost exactly.

According to some implementation forms, the statistical transformationfunction T(x) may be defined according to the following formula:

${{T(x)} = {\sqrt{\frac{\sigma_{X}^{2}}{\sigma_{X}^{2} + \frac{\Delta^{2}}{12}}}x}},$where Δ is a quantizer step size, and

σ_(X)² = ∫_(−∞)^(∞)x²f_(X)(x) 𝕕xis the variance of variable X that is related to the probability densityfunction ƒ_(X)(·) of the first probability density.

This transformation function T(x) may be used if all, for example two,descriptions are received, and the first probability distribution shouldbe preserved approximately within a third probability distribution. Inparticular, the first and the second statistical moment of the thirdprobability distribution are preserved to be equal or approximatelyequal to the first and the second statistical moment of the firstprobability density.

According to some implementation forms, the transformation function T(x)may be defined according to the following formula:

${{T(x)} = {F_{X}^{- 1}\{ {\frac{1}{2\; M\;\Delta}{\sum\limits_{i \in {{P{(M)}}{\Delta_{l}{(i)}}}}\;{\int_{\Delta_{l}{(i)}}^{\Delta_{r}{(i)}}{{F_{X}( {x - \tau} )}\ {\mathbb{d}\tau}}}}} \}}},$where Δ is a quantizer step size, M is an index assignment parameter,F_(X)(x) is the cumulative distribution function of variable X that isrelated to the probability density function ƒ_(X)(·) of the firstprobability density, as

F_(X)(x) = ∫_(−∞)^(x)f_(X)(τ) 𝕕τ,and F_(X) ⁻¹(·) denotes the inverse cumulative distribution function,P(M) is a set of indices representing the index assignment pattern ofthe index assignment scheme being used for deriving the set ofdescriptions encoding the source signal,

${{\Delta_{l}(i)} = {{{{- ( {i - \overset{\_}{P}} )}\Delta} - {\frac{\Delta}{2}\mspace{14mu}{and}\mspace{14mu}{\Delta_{r}(i)}}} = {{{- ( {i - \overset{\_}{P}} )}\Delta} + \frac{\Delta}{2}}}},{with}$$\overset{\_}{P} = {\frac{1}{2\; M}{\sum\limits_{i \in {P{(M)}}}\;{i.}}}$

If a description is received, the above transformation function T(x) canbe used to transform the reconstructed signal such that the manipulatedreconstructed signal has a third probability density being equal orapproximately equal to the first probability density.

According to some implementation forms, the transformation function T(x)may be defined according to the following formula:

${{T(x)} = {\sqrt{\frac{\sigma_{X}^{2}}{\sigma_{X}^{2} + \frac{\Delta^{2}M^{4}}{3}}}x}},$where Δ is a quantizer step size, M is an index assignment parameter and

σ_(X)² = ∫_(−∞)^(∞)x²f_(X)(x) 𝕕xis the variance of variable X that is related to the probability densityfunction ƒ_(X)(·) of the first probability density.

If only some, for example one, description is received, and the firstprobability distribution should be approximated with less computationaleffort, the above transformation function T(x) may be used. Inparticular, the first and the second statistical moment of the firstprobability density may be preserved by this formula.

The usage of the above-described transformation functions may becombined, wherein respective usage depends on the composition of thesubset of descriptions received and the desired grade of preserving orapproximation of the first probability density for the third probabilitydensity of the manipulated reconstructed signal.

According to a second aspect, the invention relates to a decoder forreconstructing a source signal, which is encoded by a set of at leasttwo descriptions, and the source signal having a first probabilitydensity, wherein the first probability density comprises a firststatistical moment and a second statistical moment. The decodercomprises an input for receiving a subset of the set of descriptions, areconstructor for reconstructing a reconstructed signal at an operatingbitrate of a set of operating bitrates upon the basis of the subset ofdescriptions, the reconstructed signal having a second probabilitydensity, wherein the second probability density comprises a firststatistical moment and a second statistical moment, and a transformerfor manipulating the reconstructed signal in order to obtain amanipulated reconstructed signal having a third probability density,wherein the third probability density comprises a first statisticalmoment and a second statistical moment, wherein the transformer isconfigured to manipulate the reconstructed signal such that at least thefirst statistical moment and the second statistical moment of the thirdprobability density are more similar to the first statistical moment andthe second statistical moment of the source signal than the firststatistical moment and the second statistical moment of the secondprobability density are, and wherein the transformer is furthermoreconfigured to manipulate the reconstructed signal such that,irrespective of the operating bitrate, a predetermined minimumsimilarity between the first statistical moment of the third probabilitydensity and the first statistical moment of the first probabilitydensity and between the second statistical moment of the thirdprobability density and the second statistical moment of the firstprobability density is maintained.

According to a first implementation form of the second aspect, theinvention relates to a decoder, wherein the transformer is configured tomanipulate the first statistical moment and the second statisticalmoment of the second probability density in order to preserve the firststatistical moment and the second statistical moment of the firstprobability density within a predetermined moment range. Hence, thetransformer can be arranged such that the predetermined moment range canbe achieved.

According to a second implementation form of the second aspect, theinvention relates to a decoder, wherein the reconstructor comprises acentral reconstruction path, which is configured to reconstruct thereconstructed signal upon the basis of index information, in particularunique index information, the central reconstruction path comprising anindexer configured to determine the index information upon the basis ofthe set of descriptions.

The descriptions of the set of descriptions can for example be used togenerate one or more index values, which form the index information. Theindex information may point to a unique value in an index assignmentscheme as described above. In particular, the central reconstructionpath may be active if all descriptions are received such that the subsetof descriptions is equal to the set of descriptions.

According to a third implementation form of the second aspect, theinvention relates to a decoder, wherein the reconstructor comprises atleast one side reconstruction path, which is configured to reconstructthe reconstructed signal upon the basis of mapping information, the atleast one side reconstruction path comprising a mapper configured todetermine the mapping information upon the basis of the descriptions ofthe subset and of a composition of descriptions in the subset.

In particular, one side reconstruction path is activated or used for agiven subset of description if one or more descriptions are lost suchthat no unique value may be determined directly. However, the remainingreceived descriptions are used to determine mapping information, whichis the basis for reconstructing the reconstructed signal. Thecomposition of the subset of descriptions may determine how to derivethe reconstructed signal from the mapping information, for example.

According to a fourth implementation form of the second aspect, theinvention relates to a decoder, wherein the transformer is configured toperform the manipulating upon the basis of a composition of descriptionsin the subset of descriptions. Hence, different manipulations may beperformed for receiving the full set of descriptions or a subset ofdescriptions. Various transformation functions may be used in thetransformer, which depend on the composition of the subset and on thegrade of distribution preservation. Various embodiments of thetransformer become apparent with respect to the explanations for thesixth implementation form of the first aspect regarding the statisticaltransformation functions.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a flowchart of a method according to an implementationform;

FIG. 2 shows a block diagram of an MDC system according to animplementation form;

FIG. 3 shows a block diagram of an MDC system according to animplementation form;

FIG. 4 shows an index assignment scheme according to an implementationform;

FIG. 5A shows an example of the side reconstruction for the case whenone description is lost according to an implementation form;

FIG. 5B shows a partition of side quantization cells into centralquantization cells; and

FIG. 6 shows an audio encoder according to an implementation form.

DETAILED DESCRIPTION OF THE EMBODIMENTS

FIG. 1 shows a flowchart of a method for reconstructing a source signalaccording to an implementation form. As a prerequisite, in step 101, thesource signal is encoded by a set of at least two descriptions, forexample by an MDC encoder. The source signal has a first probabilitydensity, which comprises a first statistical moment and a secondstatistical moment. The set of descriptions is transmitted over acommunication network, which may be unreliable, since all or somedescriptions may be lost during the transmission resulting in asituation that only a subset of the descriptions is available to thedecoder.

In step 103, a subset of the set of descriptions is received. The subsetmay be equal to the set of descriptions, such that all descriptions usedfor encoding the source signal are received. However, as thecommunication network may be unreliable, the subset may comprise asmaller number of descriptions than the full set of descriptions.

In step 105, a reconstructed signal is reconstructed at an operatingbitrate of a set of operating bitrates upon the basis of the subsets ofdescriptions that have been received in step 103. The reconstructedsignal has a second probability density, which comprises a firststatistical moment and a second statistical moment. The quality ofreconstruction may be dependent on the composition of descriptions inthe received subset of descriptions. In general, it can be assumed thatthe more descriptions are received, the better the quality of thereconstructed signal is.

In step 107, the reconstructed signal is manipulated in order tomaintain a manipulated reconstructed signal having a third probabilitydensity, which comprises a first statistical moment and a secondstatistical moment. The reconstructed signal is manipulated such that atleast the first statistical moment and the second statistical moment ofthe third probability density are more similar to the first statisticalmoment and the second statistical moment of the source signal than thefirst statistical moment and the second statistical moment of the secondprobability density are. Furthermore, the reconstructed signal ismanipulated such that, irrespective of the operating bitrate, apredetermined minimum similarity between the first statistical moment ofthe third probability density and the first statistical moment of thefirst probability density and between the second statistical moment ofthe third probability density and the second statistical moment of thefirst probability density is maintained.

The above method steps will be explained in more detail using anexemplary MDC encoding and decoding scheme, which in particular uses atwo-description encoding/decoding.

It shall, however, be noted that, the source signal may be encoded by aset of more than two descriptions. In that case, for each descriptionsub-set, a side dequantizer at the decoder is deployed. The decoder mayfurther comprise a central quantizer.

The scheme is operating on a scalar random variable that is distributedaccording to some known probability density function (p.d.f.). In thegeneral case, the quantized signal has a non-zero mean μ. The mean μ maybe subtracted before coding without affecting generality of the scheme.After subtracting the mean, we obtain a zero-mean signal represented bya random variable X that is distributed according to some known p.d.f.ƒ_(X) may be known by the encoder and the decoder. In particular,information about ƒ_(X) may be transmitted to the decoder or the decodermay use an approximation to ƒ_(X).

The variable X will be quantized by multiple description quantization.According to some scenario, the samples of the signal represented by therandom variable X are independent. In the considered example the codingis performed on per-sample basis.

The first operation comprises adding a sample of a pseudo-random ditherZ to the input sample X, the result of this operation is then quantizedby means of a uniform scalar quantizer with a step size Δ. Thequantization yields a quantization index IC. The quantization mayconveniently be implemented by means of rounding. Let [.] denote arounding operation. Therefore,I _(C)=[(X+Z)/Δ]  (1)

The encoder and the decoder may be configured in the same way, e.g. theyboth are operating with the same step size of the central quantizer A,and they use exactly the same index assignment configuration. Inparticular, this condition may be achieved when Δ and M are predefinedor transmitted to the decoder.

The randomness of the dither signal Z may be shared between the encoderand the decoder, e.g. the dither signal Z is pseudo-random. Inparticular, the dither Z may be generated by a random number generationthat facilitates generation of the random signal that is uniformlydistributed within a predefined support region. In particular, thedither signal Z may be uniformly distributed within the interval (0,2MΔ). If the pseudo-random dither is used, the synchronicity between theencoder and the decoder must be maintained. This may be achieved, forexample, when the state of the random number generation is reset on afixed time basis rendering synchronization between the encoder and thedecoder.

The index IC is a so-called central index. The central index IC may bemapped to a pair of so-called side indices I0 and I1 by means ofso-called index assignment algorithm, IA. The indices I0 and I1 may befurther encoded by means of an entropy coder, e.g. arithmetic encoder,to generate a variable length binary code.

The indices I0 and I1 are transmitted independently through anunreliable network. At this point, we call them the descriptions.Independent transmission may be achieved by embedding a bit-streamassociated with the first description into one packet and embedding thebit-stream associated with the second description into another packet.Some or all of the descriptions may be lost during the transmission. Theoperation of the decoder depends on the number and/or the composition ofthe received descriptions. The general notion is that all the receiveddescriptions are used to perform signal reconstruction. The moredescriptions are received the better the quality of the reconstruction.The possible cases of description losses are as follows.

CASE 1: It is assumed that both descriptions I0 and I1 are received.

In this case, since the decoder knows the index assignment (IA), it isable to determine uniquely the quantization index of the centralquantizer IC that corresponds to that particular pair of I0 and I1.Since the central quantizer index IC is known, a decoder of the centralquantizer Q⁻¹ may be used to perform the reconstruction. This operationis implemented according to the following formula:X _(C) =I _(C)·Δ.  (2)

In the next step, the pseudo-random dither Z is subtracted from X_(C),according to{tilde over (X)} _(C) =X _(C) −Z.  (3)

It is assumed that the decoder has access to Z, which is possible, as Zis pseudo-random.

The statistical distribution p.d.f. of {tilde over (X)}_(C) may be knownprecisely or at least approximately precise. For example, according tosome implementation forms, {tilde over (X)}_(C) may be distributed as

${f_{X}*{U( {{- \frac{\Delta}{2}},\frac{\Delta}{2}} )}},$where * denotes a convolution operator. Therefore, it is possible toderive an analytical transformation that operates on {tilde over(X)}_(C) and transforms {tilde over (X)}_(C) into a new variable {tildeover (X)}_(C) that is distributed precisely according to the same p.d.f.as X. The transformation may be of the form:

$\begin{matrix}{{{T(x)} = {F_{X}^{- 1}\{ {\frac{1}{\Delta}{\underset{- \frac{\Delta}{2}}{\int\limits^{\frac{\Delta}{2}}}{{F_{X}( {x - \tau} )}{\mathbb{d}\tau}}}} \}}},} & (4)\end{matrix}$where F_(X)(x) is the cumulative distribution function (c.d.f.) ofvariable X that is related to the p.d.f. ƒ_(X)(·), as

$\begin{matrix}{{{F_{X}(x)} = {\underset{- \infty}{\int\limits^{x}}{{f_{X}(\tau)}{\mathbb{d}\tau}}}},} & (5)\end{matrix}$and F_(X) ⁻¹(·) denotes the inverse c.d.f. It is assumed that F_(X)⁻¹(α)=inf_(β){F_(X)(β)≧α}. Therefore, the new variable {circumflex over(X)}_(C) is computed according to{circumflex over (X)} _(C) =T({tilde over (X)} _(C)).  (6)

The last decoding stage may comprise adding back the mean value vtyielding a final reconstruction.

In some implementations, it may be of practical interest to relax thedistribution preserving constraint to a setting where the distributionis preserved approximately. This could be seen as a low complexityalternative of the main algorithm. In the case of preserving thedistribution approximately, the transformation may be of the form:

$\begin{matrix}{{{T(x)} = {\sqrt{\frac{\sigma_{X}^{2}}{\sigma_{X}^{2} + \frac{\Delta^{2}}{12}}}x}},} & {(7),}\end{matrix}$where

$\sigma_{X}^{2} = {\underset{- \infty}{\int\limits^{\infty}}{x^{2}{f_{X}(x)}{\mathbb{d}x}}}$is the variance of X.

CASE 2: It is assumed that one description is received.

If one description is received, the side dequantizer associated with thedescription that was successfully received is used. Suppose that onedescription was lost and only the index Ij is available to one decoder(j=0 or 1). The index Ij is uniquely associated, e.g. by means ofone-to-one mapping, with some reconstruction point of the j-th sidequantizer that we denote by {tilde over (X)}′_(j). The reconstructionpoints of the side quantizers may be pre-computed and the mapping formIj to {tilde over (X)}′_(j) may be implemented by means of a look-uptable. An explanation what the reconstruction points are and how theyare related to the index assignment will be described later inconjunction with FIG. 4 and FIG. 5A.

The intermediate reconstruction is obtained by selecting areconstruction point {tilde over (X)}′_(j) of the j-th side dequantizerthat is associated with the received index Ij and subtracting therealization of the pseudorandom dither Z, i.e.{tilde over (X)} _(j) ={tilde over (X)}′ _(j) −Z.  (8)

One property of the proposed approach is that the statisticaldistribution of {tilde over (X)}_(j) is essentially known in ananalytical form. For example, {tilde over (X)}_(j) may be distributed asƒ_(X)*ƒ_(N) where * denotes a convolution operator and ƒ_(N) is theprobability density function of the quantization noise of the j-th sidequantizer. The probability density function of the quantization noiseƒ_(N), according to some implementation forms, may be uniform within aside quantization cell specified by P(M) and A that it is reflectedaround its origin P·Δ, i.e.,

${f_{N} = {\frac{1}{2\; M\;\Delta}{\sum\limits_{i \in {P{(M)}}}^{\;}\;{U( {{\Delta_{l}(i)},{\Delta_{r}(i)}} )}}}},$where Δ_(i)(i), Δ_(r)(i) and P are to be defined with the followingequation (9).

The form of the statistical distribution is analytical and thus itfacilitates derivation of an analytical distribution preservingtransformation. In this case the distribution preserving transformationmay be computed as:

$\begin{matrix}{{{T_{j}(x)} = {F_{X}^{- 1}\{ {\frac{1}{2\; M\;\Delta}{\sum\limits_{i \in {P{(M)}}}^{\;}\;{\underset{\Delta_{l}{(i)}}{\int\limits^{\Delta_{r}{(i)}}}{{F_{X}( {x - \tau} )}{\mathbb{d}\tau}}}}} \}}},} & (9)\end{matrix}$where the integration boundaries and Δ_(l)(i) and Δ_(r)(i) may becomputed as follows. In order to compute Δ_(l)(i) and Δ_(r)(i), P needsto be defined, namely the mean of an index pattern specified by some setP(M). F_(X)(x) is the c.d.f. of variable X that is related to the p.d.f.and F_(X) ⁻¹(·) denotes the inverse c.d.f.

An exemplary index assignment scheme may have a property that itgenerates always the same pattern of the central cells within each sidecell. This property facilitates dithering and obtaining an analyticaladditive noise model of side quantizer that is accurate at any bitrateand any configuration of the index assignment, specified by theparameter M. It is assumed, for example, that the pattern of centralindices within any side quantization cell is specified by the followingset:P(M)={0}∪{i(M−1)}_(i=1) ^(M-1)∪{(M−1)² +M}∪{(M−1)² +M+i(M+1)}_(i=1)^(M-1).  (10)

It can be seen that the set P(M) contains precisely 2M elements,therefore every side quantization cell contains precisely 2M centralcells. The expression (10) may be used for any M, however, lower valuesof M may be of more practical relevance. The parameter M facilitates atrade-off between the distortion of the central quantizer and thedistortion of the side quantizer.

It is assumed, for example, that the decoder knows the configuration ofthe index assignment that was used during the encoding, specified by M,therefore, it is able to compute P(M). For instance, let M=2. Thecorresponding pattern of indices is P(2)={0,1,3,6}.

P may be computed as

$\begin{matrix}{\overset{\_}{P} = {\frac{1}{2M}{\sum\limits_{i \in {P{(M)}}}^{\;}\;{i.}}}} & (11)\end{matrix}$

The integration boundaries Δ_(l)(i) may then be computed as

$\begin{matrix}{{{\Delta_{l}(i)} = {{{- ( {i - \overset{\_}{P}} )}\Delta} - \frac{\Delta}{2}}},{and}} & (12) \\{{\Delta_{r}(i)} = {{{- ( {i - \overset{\_}{P}} )}\Delta} + {\frac{\Delta}{2}.}}} & (13)\end{matrix}$

Note that the integration boundaries may be pre-computed for any given Mand Δ leading to an efficient implementation. The transformation T_(j)is applied to {tilde over (X)}_(j) yielding new variable {circumflexover (X)}_(j)=T_(j)({tilde over (X)}_(j)) that is distributed accordingto the p.d.f. of the input signal X. As a final decoding step, the meanvalue μ may be added back to the reconstructed signal yielding a finalreconstruction.

In some cases, it may be desired to use a low complexity alternative ofthe distribution preserving transformation. Instead of preserving theoriginal source p.d.f. exactly, it is possible to preserve itapproximately by preserving only the first and the second statisticalmoment of ƒ_(X), the mean and the variance, respectively. The periodicpattern of central quantization indices within the side quantizationcell facilitates derivation of a moment preserving transformation. Thetransformation is of form

$\begin{matrix}{{{T_{j - {L\; C}}(x)} = {\sqrt{\frac{\sigma_{X}^{2}}{\sigma_{X}^{2} + \sigma_{j}^{2}}}x}},} & (14)\end{matrix}$where σ_(j) ² is the quantization noise error variance for the side j-thquantizer, which is a function of Δ and M. The value of σ_(j) ² may becomputed from the following formula:

$\begin{matrix}{\sigma_{j}^{2} = {\frac{\Delta^{2}M^{4}}{3}.}} & (15)\end{matrix}$

CASE 3: It is assumed that no description is received.

If no description is received, the decoder still performs signalreconstruction. In this case the reconstruction is obtained by means ofsignal synthesis utilizing the available information about the signalp.d.f. ƒ_(X) or its approximation. There are several state-of-the-artmethods that may be used to perform synthesis. In particular, theinverse sampling method leads to a neat construction of the decoder forthe case if all the descriptions are lost. Assuming that thesynchronization between the encoder and decoder is maintained, thedecoder has access to the dither signal Z˜U{0,2MΔ}. In this case, areconstruction {circumflex over (X)}_(Z) may be received by applying thefollowing transformation directly to the dither signal:

$\begin{matrix}{{{T_{Z}(z)} = {F_{X}^{- 1}( \frac{z}{2\; M\;\Delta} )}},} & (16)\end{matrix}$where F_(X) ⁻¹(·) denotes the inverse c.d.f. It is assumed that F_(X)⁻¹(α)=inf_(β){F_(X)(β)≧α}. The reconstruction {circumflex over (X)}_(Z)is then obtained by taking {circumflex over (X)}_(Z)=T_(Z)(Z). As afinal decoding step, the mean value μ may be added to {circumflex over(X)}_(Z) yielding a final reconstruction.

In some cases, it might be of practical interest to consider a lowcomplexity alternative to the usage of T_(Z)(·). If the distributionpreserving constraint is relaxed and the distribution is preservedapproximately, the reconstruction may be obtained by using the followingtransformation that preserves the second moment of the original densityƒ_(X):

$\begin{matrix}{{T_{Z - {L\; C}}(z)} = {\sqrt{\frac{12\;\sigma_{x}^{2}}{\Delta^{2}}} \cdot {( {z - {M\;\Delta}} ).}}} & (17)\end{matrix}$

Regarding the operating bitrate during reconstruction, it may bereferred to as the total bitrate that is distributed among thedescriptions, e.g. the bitrate available to the quantizer. In general,the bitrate may be distributed arbitrarily between the descriptions.According to some embodiments, the allocation of bitrate among thedescriptions is symmetrical, meaning that all the descriptions areallocated with the same bitrate.

The bitrate may depend on the particular selection of Δ, M and the firstprobability density function ƒ_(X).

The total bitrate may be subject to frequent changes because ƒ_(X) maychange in time. In addition, some properties of the communicationchannel may vary during the transmission, e.g. due to packet loss rateand/or available transmission bitrate. These changes affect values of Δand M influencing the operating bitrate.

Regarding similarity between the third and the first probabilitydensity, in various implementations it is aimed at preserving the firstp.d.f., meaning that the reconstructed signal may be distributedaccording to the first p.d.f. This approach can be motivated ontheoretical grounds and also it is experimentally proven to beperceptually efficient. However, the benefit could also be obtained ifthe first p.d.f. is preserved approximately by allowing a smalldeviation from the first p.d.f. There are several similarity measuresthat may be considered.

The simplest measure may be an absolute difference between thestatistical moments. Let us consider two probability distributionfunctions ƒ_(X) and ƒ_(Y). Both p.d.f.s comprise their first and secondstatistical moments. The first statistical moment is the mean. In thiscase we have:

$\begin{matrix}{{\mu_{X} = {\underset{- \infty}{\int\limits^{\infty}}{{{xf}_{X}(x)}{\mathbb{d}x}}}}{and}} & (18) \\{\mu_{Y} = {\underset{- \infty}{\int\limits^{\infty}}{{{xf}_{Y}(x)}{{\mathbb{d}x}.}}}} & (19)\end{matrix}$

The second statistical moment is the variance. We define

$\begin{matrix}{{\sigma_{X}^{2} = {\underset{- \infty}{\int\limits^{\infty}}{( {x - \mu_{X}} )^{2}{f_{X}(x)}{\mathbb{d}x}}}}{and}} & (20) \\{\sigma_{Y}^{2} = {\underset{- \infty}{\int\limits^{\infty}}{( {x - \mu_{Y}} )^{2}{f_{Y}(x)}{{\mathbb{d}x}.}}}} & (21)\end{matrix}$

The absolute difference similarity measure may be defined for the firstmoment asη_(μ)=|μ_(X)−μ_(Y)|,  (22)and for the second moment asη_(σ) ²=|σ_(X) ²−σ_(Y) ²|.  (23)

According to some implementations, it may be more convenient to define anormalized absolute difference similarity measure. In this case, thesimilarity between the first moment may be defined as

$\begin{matrix}{{\gamma_{\mu} = \frac{{\mu_{X} - \mu_{Y}}}{\mu_{X}}},} & (24)\end{matrix}$and for the second moment

$\begin{matrix}{\gamma_{\sigma^{2}} = {\frac{{\sigma_{X}^{2} - \sigma_{Y}^{2}}}{\sigma_{X}^{2}}.}} & (25)\end{matrix}$

If the normalized absolute difference similarity measure is used, aninterval may be formulated where the range of similarity is defined. Forexample, the following thresholds may be set up:γ_(σ) ₂ ≦½,γ_(μ)≦½.  (26)

In general, any deviations from the original moments may not be allowed,but it may be chosen that a 3 dB difference is acceptable, which wouldlead to 50% energy loss/increase of the signal. For example, assumingthat at most a 3 dB difference is acceptable, for the second moment, thefollowing test formula may be used:

$\begin{matrix}{\Gamma_{\sigma^{2}} = \{ {\begin{matrix}{{10\;\log_{10}\beta},} & {\beta \geq 1} \\{{10{\log_{10}( {\beta^{- 1} + \frac{1}{2}} )}},} & {\beta < 1}\end{matrix},{{{with}\mspace{14mu}\beta} = \frac{\sigma_{X}^{2}}{\sigma_{Y}^{2}}},} } & (27)\end{matrix}$it may be defined that Γ_(σ) ₂ ≦3. Thus, the above equation provides atest whether the 3 dB condition is fulfilled or not.

According to some implementation forms, the predetermined minimumsimilarity may be the same for the first and the second statisticalmoments. However, a predetermined minimum similarity may be chosen suchthat less difference is allowed for the first statistical moments thanfor the second statistical moments. In particular, none or only anegligible difference may be allowed for the first statistical moments,while a certain difference, e.g. 3 dB, may be allowed for the secondstatistical moments.

For example, assuming that at most a 3 dB difference is acceptable, forthe first moment, the following test formula may be used:

$\begin{matrix}{\Gamma_{\mu} = \{ {\begin{matrix}{{10\;\log_{10}\beta},} & {\beta \geq 1} \\{{10{\log_{10}( {\beta^{- 1} + \frac{1}{2}} )}},} & {\beta < 1}\end{matrix},{{{with}\mspace{14mu}\beta} = {{\frac{\mu_{X}}{\mu_{Y}}}.}}} } & (28)\end{matrix}$

The allowed deviation may be therefore expressed by the followingconstraint Γ_(μ)≦3 in particular it may be desirable to have Γ_(μ)=0.

Since the probability density functions considered here always comprisethe first and the second statistical moment, the proposed similaritymeasure will work also in the case when the distributions are preservedexactly, i.e. full complexity case. Alternatively, in the case ofpreserving of a p.d.f. exactly, one may use the Kullback-Leiblerdivergence.

The Kullback-Leibler divergence between ƒ_(X) and ƒ_(Y) is defined as

$\begin{matrix}{D_{K\; L}( {{{f_{X} f_{Y} )} = {\underset{- \infty}{\int\limits^{\infty}}{{f_{X}(x)}\log_{2}\frac{f_{X}(x)}{f_{Y}(x)}{\mathbb{d}x}}}},} } & (29)\end{matrix}$which is measured in bits. This measure is particularly useful toevaluate the case where the third distribution is intended to beessentially exactly the same as the first distribution.

FIG. 2 shows a block diagram of an MDC system according to animplementation form. Although the main focus of the description isdirected to the decoding part of the MDC system, an encoding part isdescribed as well for comprehensive understanding.

In block 201, a source signal is provided, wherein a statistical signalmodel of the source signal to be quantized is known. For example, such astatistical signal model may be represented or be equivalent to thep.d.f., which may be called the first p.d.f. The source signal isprovided at block 203 that, for example, incorporates a multipledescription quantizer (MDQ). The quantizer may use dithering and aperiodic index assignment, as described above, in order to generate aset 205 of indices or descriptions encoding the source signal. The set205 of descriptions is transmitted to a decoder side, which includes areconstructor 207, a transformer 209 and an output buffer 211. Duringtransmission, none, some or all descriptions may be lost, such that asubset of descriptions is received at the decoder side. The subset ofdescriptions accordingly may be equal to the set of descriptions or be areceived subset of descriptions or be an empty set.

Depending on the composition of the subset of descriptions received,various distortion levels 1, . . . , N, . . . , L may be present at thereconstructor 207. For example, in a distortion level 1, all thedescriptions are received. In a distortion level N, for example, N outof M descriptions are received, and in the distortion level L, all thedescriptions are lost. The reconstructor 207 comprises severalreconstructor blocks 213, 215, 217, which are respectively adapted toreconstruct a reconstructed signal based on the specific distortionlevel defined by the composition of the subset of descriptions received.

Due to the reconstruction, a respective reconstructed signal is providedto the transformer 209, the respective reconstructed signal having asecond p.d.f. that is more or less similar to the first p.d.f., thesimilarity depending on the composition of the subset. However, thestatistical properties of the reconstructed signal may be known inadvance depending on the respective distortion level. Accordingly, inprocessing blocks 219, 221, 223 of the transformer 209, thereconstructed signal having the second p.d.f. can be statisticallymanipulated by means of a transformation function, which achieves athird p.d.f. for the manipulated reconstructed signal, which may be are-establishment of the first p.d.f. Regarding various implementationforms of the transformation functions, it is referred to equations (4)to (17). The respective output of the transformer 209 is provided to thebuffer 211, which comprises blocks 225, 227, 229 for the respectivestatistically manipulated reconstructed signal of the correspondingdistortion level, each having a probability density equal orapproximately equal to the first p.d.f., as described above.

FIG. 3 shows a further implementation form of an MDC system, wherein aninput source signal 301 is encoded in an encoder 303 and decoded in adecoder 305. In the encoder 303, which is explained briefly for acomprehensive understanding of the decoder 305, the dither Z is added tothe source signal 301 with an adder 307. The output of the adder 307 isprovided to the quantizer 309 and further to an index assignment block311, which generates two descriptions I0, I1, which may then betransmitted to the decoder 305. The function of the encoder 303 may beexpressed by equation (1).

The decoder 305 may receive both descriptions I0, I1, one of thedescriptions I0 or I1 or none of the descriptions. If both descriptionsI0, I1 are received, they are provided to a reverse index assignmentblock 313, which performs a mapping of the received indices onto a valueor a set of values, which is provided to a dequantizer 315. Theoperation of the dequantizer 315 may be implemented by equation (2). Theresult of the dequantizer 315 is provided to an adder 317, wherein thedither Z is subtracted, e.g. according to equation (3). The resultingvalue is provided to a central transformer 319, which may operateaccording to equations (4) or (7). Hence, in block 321 a manipulatedreconstructed signal is available having a p.d.f. being the same as orsimilar to the first p.d.f. of the signal source. Blocks 313, 315, 317,319, 321 form a central reconstruction path.

If only one of the descriptions I0, I1 is received, it is provided to arespective dequantizer and reconstructor 323 or 325, respectively. Inthese blocks 323, 325, a reconstruction point is determined from thereceived description, which will be explained in more detail withrespect to FIG. 4 and FIG. 5A. Similar to adder 317, the dither signal Zis subtracted from the reconstruction points at adders 327 or 329,respectively, for example according to equation (8). The resultingsignal is provided to transformation blocks 331 or 333, respectively,which perform a statistical manipulation or transformation of thesignals, e.g. according to equations (9) or (14).

The resulting manipulated reconstructed signal is then provided toblocks 335 or 337, respectively, such that the reconstructed outputsignals have a third p.d.f. being equal or approximately equal to thefirst p.d.f. of the source signal.

Blocks 323, 327, 331 and 335 form a first side reconstruction path, andblocks 325, 329, 333, 337 form a second side reconstruction path.

As shown in FIG. 3, the side quantizer 0 comprises blocks 309, 311 andthe output I0. The side quantizer 1 comprises blocks 309, 311 and theoutput I1.

A similarity between the probability densities of the signals in blocks321, 335, 337 and the first probability density of the source signal maybe defined in advance according to equations (18) to (29).

If all descriptions are lost, i.e. no description is received, areconstructed signal may be synthesized in a synthesizer block 339,which, for example, operates according to equation (16) or (17). Inparticular, synthesis may be based on the dither signal Z. In the caseof synthesis, a synthesized reconstruction signal may be provided inblock 341, the reconstructed signal having a probability density beingequal to or approximately equal to the first p.d.f. of the sourcesignal.

For the purposes of better illustrating the operation of the decoder305, a particular configuration of the index assignment algorithm may bechosen. According to the above embodiment, the parameter M is chosen toM=2, therefore the index assignment matrix may have four diagonals. Anexemplary index assignment scheme or index assignment matrix is shown inFIG. 4. Note that the index assignment generates the same pattern of acentral cell within each side cell. The pattern is described by a setP(M), which in this case is P(2)={0, 1, 3, 6}. The pattern may beobtained by selecting a particular side quantization cell andsubtracting the smallest central cell index within that side cell fromall central indices belonging to that cell. For example, if side cell 2of a side quantizer 0 (402) is considered, the central cells belongingto that side cell are {4, 5, 7, 10}, and the pattern P(2) may becomputed as P(2)={4-4, 5-4, 7-4, 10-4}={0, 1, 3, 6}. For example, ifside cell 3 of side quantizer 1 (413) is considered, the central cellbelonging to that side cell are specified by a set {6,7,9,12}. Thecorresponding pattern is therefore P(2)={6-6, 7-6, 9-6, 12-6}={0, 1, 3,6}.

The columns of the index assignment matrix shown in FIG. 4 are indexedby indices of the side quantizer 0 and the rows are indexed by indicesof the side quantizer 1. The indexing of the side quantization case maybe done in an arbitrary manner. However, each side quantization cell ofeach side quantizer may be indexed by a unique index. In the exampleshown in FIG. 4, it is assumed that the central index 7 was transmitted.The index is mapped to a pair of indices. In this case, index 2 of theside quantizer 0 is defined by column 402, and index 3 of the sidequantizer 1, defined by row 413. It is referred to the indices in such apair as side indices. If both side indices are available, the indexassignment may be inverted and a corresponding central index may bedetermined uniquely from the index assignment matrix. As described, someside indices may be lost during the transmission. In this case, onlythese side indices that are available to the decoder are used for thereconstruction.

For the purposes of the illustration, we assume that the side index ofthe side quantization 0 has been lost. This implies that only the sideindex of the side quantizer 1 is available to the decoder. In this case,it is the index 3. It is known that the side index 3 is associated witha particular side quantization cell of row 413 that is a disjoint unionof central cells. The side quantization cell consists of the followingcentral cells {6, 7, 9, 12}. In this case, the pattern P(M) describingthe corresponding configuration of the index assignment is {0, 1, 3, 6}.The pattern may be obtained by subtracting the smallest central indexbelonging to the side quantization cell from all the indices of thecentral cells belonging to the side cell. For example, P(M)={6-6, 7-6,9-6, 12-6}={0, 1, 3, 6}.

This side quantization cell 413 is shown in FIG. 5A. The sidequantization cell 413 consists of four central cells, each of which hasa width equal to the quantizer step size Δ. Each side quantization cellis associated with a single reconstruction point. The reconstructionpoint for the side quantization cell is computed by averaging thecentral indices belonging to this specific side quantization cell and byscaling the result with respect to the central quantizer step size Δ. Inthis specific case, the mean value of the pattern is equal to8.5=(6+7+9+12)/4. The exact position of the reconstruction point istherefore 8.5·Δ. It may be shown that such a method to compute theposition of the reconstruction point is optimal with respect to MSE forthe considered quantizer. The position of the reconstruction pointfacilitates a reconstruction that is as close as possible, in particularin terms of MSE, to the signal sample to be quantized.

With reference to FIG. 5A, the side dequantizer performs a mappingbetween a side index or set of indices to a single reconstruction value.FIG. 5A depicts the case where the side quantization index 3 of a sidequantizer 1, as depicted in FIG. 4, is received. The index indicates theparticular side quantization cell (413) that is associated with a singlereconstruction value. The dequantizer mapping assigns the reconstructionvalue to the index or set of indices of the side quantizer.

FIG. 5B shows a partition of side quantization cells into centralquantization cells, with central quantization indices 501, sidequantization cell indices 503 of the first quantizer side, which is onlypartly shown, cell 0 of the side quantizer 1, 505, side cell 1 of theside quantizer 1, 507, side cell 2 of the side quantizer 1, 509, sidecell 3 of the side quantizer 1, 511, and side cell 4 of the sidequantizer 1, 513, which is only partly shown.

With reference to FIGS. 3 to 5B, the side quantizer uses the partitionof the central quantizer. The use of the partition of the centralquantizer is described by the index assignment algorithm. Allquantization cells of the central quantizer are indexed by the centralindices that are the non-empty elements, i.e. elements having entries,of the index assignment matrix shown in FIG. 4. All central quantizationcells are also indexed by side quantization indices of each sidequantizer that is used.

The quantization cells {6, 7, 9, 12} of the side quantizer 1, denoted byreference sign 413 in FIG. 4 can for example be associated with sideindex 3 of the side quantizer 1. This means that they belong to a singlequantization cell of the side quantizer 1. This cell is shown in FIG.5A.

The central cells associated with indices {6, 7, 9, 12} shown in FIG. 4belong for example to different cells of the side quantizer 0. Forexample, the central cell 6 belongs to the side quantization cell 1 ofthe side quantizer 0, the central cell 7 belongs to the sidequantization cell 2 of the side quantizer 0, and so forth.

The sample that was quantized in order to be assigned to this specificquantization cell must have had a value within one of the intervals

$\lbrack {{{6\;\Delta} - \frac{\Delta}{2}},{{6\;\Delta} + \frac{\Delta}{2}}} \rbrack,\lbrack {{{7\;\Delta} - \frac{\Delta}{2}},{{7\;\Delta} + \frac{\Delta}{2}}} \rbrack,{\lbrack {{{9\Delta} - \frac{\Delta}{2}},{{9\Delta} + \frac{\Delta}{2}}} \rbrack\mspace{20mu}{{{and}\;\lbrack {{{12\;\Delta} - \frac{\Delta}{2}},{{12\;\Delta} + \frac{\Delta}{2}}} \rbrack}.}}$

If the other description is not available, the decoder has no means touniquely determine the true interval. It can, however, use areconstruction point that leads to the minimum possible performance lossbased on the mentioned intervals. Since there are more central cells atthe left side of the considered side quantization cell, it is morelikely that the true value of the signal to be quantized is locatedthere. For example, it is the case where the reconstruction point islocated in the position equal to the mean of the pattern of indices inthe respective side quantization cell. It turns out that the bestposition of the reconstruction point in this case may be 8.5·Δ.

The exact positions of the reconstruction points may be found similarlyfor the other quantization cells and the other side quantizer. Themapping from the side quantizer index to the side quantizerreconstruction point can be pre-computed during the design of themultiple description quantizer. It may be conveniently implemented usinga look-up table or, in the case of the specific index assignment that isused in this example, by rounding.

It may be noticed from the above example that usage of a large number ofdiagonals in the index assignment matrix leads to the situation wherethe side quantization cells are wide and thus the reconstruction withthe side quantizers is coarse. By adjusting the number of diagonals inthe index assignment matrix it is possible to perform a trade-offbetween the central and the side distortion. Usually this is done tooptimize the MSE performance of the quantizer with respect to thedescription loss probability.

According to the above-described implementation forms of the decodingmethod and the decoder, a preservation or an approximate preservation ofthe signal distribution leads to improving a perceptual performance e.g.of multimedia communications employing MDC. The distribution preservingmultiple description quantization described above may be particularlyefficient in low bitrate coding scenarios, which may be of highpractical relevance. In particular for those low bitrates, the describedimplementation forms provide a better performance than conventionalMSE-based decoders. Furthermore, a continuous transition betweenparametric descriptions and rate distortion optimal descriptions in afunction of the available bitrate may be possible. A flexibility ofcoding may be maintained such that, in particular, the encoder formultiple description coding may be redesigned and optimized based on ananalytic criterion without the need of iterative or training procedureson the encoder and the decoder sides.

The above-described coding and decoding schemes may, for example, beused for quantization in multimedia communication systems overunreliable networks, and in particular for applications in speech,audio, image or video transmission, respectively. Furthermore, thedescribed implementation forms can be used to extend a bandwidth ifapplied to multiple description coding of speech signals or audiosignals. Due to the adaptability of the decoding scheme, a flexiblemultiple description quantization of a speech signal may be achieved,where the encoder may have a vocoder-like behavior at low operatingbitrates and perform a rate distortion optimal coding for higheroperation bitrates.

The proposed multiple description distribution preserving quantization(MD-DPQ) provides means for packet-loss concealment (PLC) in multimediacommunications over communication networks with packet losses. Inparticular, it is useful in the situation when the coding is constrainedby a delay constraint that prevents form using approaches involvingforward error correction (FEC). In addition to that, the proposed MD-DPQis particularly applicable in the situations where the exact packet lossprobability remains unknown, which prevents from designing an efficientFEC coding strategy. The MD-DPQ is still operational also in thesituation when the packet-loss rate is different than the packet-lossrate assumed in the design.

The main advantage of the proposed MD-DPQ over the state-of-the-artmultiple description quantization is related to the fact that it usesmean squared error criterion combined with a distribution preservingconstraint that is perceptually more efficient than relying exclusivelyon the mean squared error in the case of the state-of-the-art systems.Another advantage of the proposed method is that it can be optimized onthe fly as it does not employ any iterative procedures. Due to thisproperty, it may be used in a context of rate-distortion flexible codingwhere it is important to be able to redesign the quantizers to match anybitrate constraint. The flexible coding is useful, for instance, if theproperties of communication channels are variable, e.g. availablebandwidth, bit-rate, packet-loss rate.

An exemplary application of the proposed MD-DPQ is in real-time audiocommunications over a network with packet losses. It is proposed toapply MD-DPQ in a forward-adaptive flexible coder that operates on monowide-band signal. A high-level block diagram of such an audio encoder600 is shown in FIG. 6.

The audio encoder 600 comprises a signal modeling block 601, aperceptual weighting block 603, a Karhunen-Loeve transformer (KLT) 605,a normalizer 607, an MD-DPQ 609, an entropy coder 611, a de-normalizer613, an inverse KLT 615, a perceptual un-weighting block 617 and apredictor 619.

Coding is performed in a forward adaptive manner. Each block of thesignal is modeled by a multivariate Gaussian in block 601. Thecovariance matrix is parameterized with an aid of an autoregressive (AR)model. Each signal block of 20 millisecond (ms) is subdivided into foursub-blocks and the AR model is interpolated for each sub-block. The ARmodel is used to obtain perceptual weighting and un-weighting filters.

A Karhunen-Loeve transform is computed on a sub-block basis in block605. The KLT is applied to the perceptually weighted signal, which isprovided by block 603, with a subtracted prediction, which is providedby predictor 619. The KLT coefficients are normalized in block 607 andquantized in block 609 using MD-DPQ yielding a block of reconstructedvalues of the central quantizer, shown as “cd”, and two sets of indicesfor the two descriptions, shown as “sd0” and “sd1”. The indices areencoded using two separate instances of the arithmetic entropy coder 611and embedded into two packets. Each packet contains full informationabout the AR model and gains for signal normalization.

Only the reconstruction obtained from the central quantizer is used atthe encoder to close the prediction loop provided by predictor 619.Local reconstruction is performed in blocks 613, 615 and 617, which, ingeneral, provide an inverse operation to the blocks 603, 605, 607.

The decoder comprises the MD-DPQ decoder that provides signalreconstruction depending on the number of received descriptions, asdescribed above. Accordingly, if both descriptions are received, thecentral reconstruction is performed. If one of the descriptions is lost,an appropriate side decoder is selected and a reconstruction isperformed. If all the descriptions are lost, a statistical signal modelfor the previously received block is used to perform signal synthesis.

A heuristically decay factor may be chosen for the gains, e.g. 0.95,that renders a fade-out of the signal if the transmission isinterrupted. The coding is performed on signal frames e.g. having alength of 20 ms. Every 20 ms, two side descriptions are generated. Theside descriptions are two-stage. One stage contains full modelinformation, e.g. linear prediction coefficients (LPC; used to representthe AR model) and four signal gains. The other stage contains signaldescription obtained from MD-DPQ. The bit-stream of the side descriptionis obtained from an entropy coder 611.

A constant step size Δ is used for all transform coefficients, which canbe motivated to be optimal by a high-rate argument. Each two-stage sidedescription is transmitted using a single packet, and for a single frameof a signal, two packets are generated. The construction of packetsleads to duplication of model information. The total available bitratemay be distributed symmetrically between the two packets.

The audio encoder 600 with MD-DPQ is flexible and can operate at anybitrate. It means that for any total bitrate constraint, the availablebitrate may be distributed between the model and the signal part of thedescription. A constant model bitrate may be used, which is a desirablestrategy for the flexible multiple description quantization.

What is claimed is:
 1. A method for reconstructing a source signal,which is encoded by a set of at least two descriptions, the sourcesignal having a first probability density, wherein the first probabilitydensity comprises a first statistical moment and a second statisticalmoment of the first probability density, the method comprising:receiving a subset of the set of descriptions; reconstructing areconstructed signal at an operating bitrate of a set of operatingbitrates upon the basis of the subset of descriptions, the reconstructedsignal having a second probability density, wherein the secondprobability density comprises a first statistical moment and a secondstatistical moment of the second probability density; and manipulatingthe reconstructed signal in order to obtain a manipulated reconstructedsignal having a third probability density, wherein the third probabilitydensity comprises a first statistical moment and a second statisticalmoment of the third probability density, wherein the reconstructedsignal is manipulated such that at least the first statistical momentand the second statistical moment of the third probability density aremore similar to the first statistical moment and the second statisticalmoment of the first probability density than the first statisticalmoment and the second statistical moment of the second probabilitydensity are, and wherein the reconstructed signal is manipulated suchthat, irrespective of the operating bitrate, a predetermined minimumsimilarity between the first statistical moment of the third probabilitydensity and the first statistical moment of the first probabilitydensity and between the second statistical moment of the thirdprobability density and the second statistical moment of the firstprobability density is maintained.
 2. The method according to claim 1,wherein the first statistical moment and the second statistical momentof the second probability density are manipulated to preserve the firststatistical moment and the second statistical moment of the firstprobability density within a predetermined moment range.
 3. The methodaccording to claim 1, wherein the first statistical moments of the firstprobability density and the third probability density are about equal,and wherein the second statistical moments of the first probabilitydensity and the third probability density are about equal.
 4. The methodaccording to claim 1, wherein the reconstructed signal is reconstructedusing a reconstruction function that is dependent on a composition ofdescriptions in the subset of descriptions.
 5. The method according toclaim 1, wherein the source signal comprises an additive dither signal,and wherein the reconstructing comprises subtracting the dither signalfrom the reconstructed signal.
 6. The method according to claim 1,wherein the source signal comprises a pseudorandom dither signal, andwherein the reconstructing comprises subtracting the pesudorandom dithersignal from the reconstructed signal.
 7. The method according to claim1, wherein the reconstructing comprises using an index assignmentscheme, which is addressed by the descriptions of the set ofdescriptions, the index assignment scheme being used for deriving theset of descriptions encoding the source signal.
 8. The method accordingto claim 1, wherein the manipulating the reconstructed signal comprisestransforming the reconstructed signal according to a statisticaltransformation function, the transformation function being dependent ona composition of descriptions in the subset of descriptions.
 9. Themethod according to claim 1, wherein the manipulating the reconstructedsignal comprises transforming the reconstructed signal according to astatistical transformation function T(x), the transformation functionT(x) being defined according to the following formula:${{T(x)} = {F_{X}^{- 1}\{ {\frac{1}{\Delta}{\underset{\frac{\Delta}{2}}{\int\limits^{\frac{\Delta}{2}}}{{F_{X}( {x - \tau} )}{\mathbb{d}\tau}}}} \}}},$where Δ is a quantizer step size, F_(X)(x) is the cumulativedistribution function of variable X that is related to the probabilitydensity function ƒ_(X)(·) of the first probability density, as${{F_{X}(x)} = {\underset{- \infty}{\int\limits^{x}}{{f_{X}(\tau)}{\mathbb{d}\tau}}}},$and F_(X) ⁻¹(·) denotes the inverse cumulative distribution function.10. The method according to claim 1, wherein the manipulating thereconstructed signal comprises transforming the reconstructed signalaccording to a statistical transformation function T(x), thetransformation function T(x) being defined according to the followingformula:${{T(x)} = {\sqrt{\frac{\sigma_{X}^{2}}{\sigma_{X}^{2} + \frac{\Delta^{2}}{12}}}x}},$where Δ is a quantizer step size, and$\sigma_{X}^{2} = {\underset{- \infty}{\int\limits^{\infty}}{x^{2}{f_{X}(x)}{\mathbb{d}x}}}$is the variance of variable X that is related to the probability densityfunction ƒ_(X)(·) of the first probability density.
 11. The methodaccording to claim 1, wherein the manipulating the reconstructed signalcomprises transforming the reconstructed signal according to astatistical transformation function T(x), the transformation functionT(x) being defined according to the following formula:${{T(x)} = {F_{X}^{- 1}\{ {\frac{1}{2\; M\;\Delta_{\;}}{\sum\limits_{i \in {P{(M)}}}^{\;}\;{\underset{\Delta_{l}{(i)}}{\int\limits^{\Delta_{r}{(i)}}}{{F_{X}( {x - \tau} )}{\mathbb{d}\tau}}}}} \}}},$where Δ is a quantizer step size, M is an index assignment parameter,F_(X)(x) is the cumulative distribution function of variable X that isrelated to the probability density function ƒ_(X)(·) of the firstprobability density, as${{F_{X}(x)} = {\underset{- \infty}{\int\limits^{x}}{{f_{X}(\tau)}{\mathbb{d}\tau}}}},$F_(X) ⁻¹(·) denotes the inverse cumulative distribution function, andP(M) is an index assignment pattern of an index assignment scheme beingused for deriving the set of descriptions encoding the source signal,${{\Delta_{l}(i)} = {{{{- ( {i - \overset{\_}{P}} )}\Delta} - {\frac{\Delta}{2}\mspace{14mu}{and}\mspace{14mu}{\Delta_{r}(i)}}} = {{{- ( {i - \overset{\_}{P}} )}\Delta} + \frac{\Delta}{2}}}},{{{with}\mspace{14mu}\overset{\_}{P}} = {\frac{1}{2M}{\sum\limits_{i \in {P{(M)}}}^{\;}\;{i.}}}}$12. The method according to claim 1 wherein the manipulating thereconstructed signal comprises transforming the reconstructed signalaccording to a statistical transformation function T(x), thetransformation function T(x) being defined according to the followingformula:${{T(x)} = {\sqrt{\frac{\sigma_{X}^{2}}{\sigma_{X}^{2} + \frac{\Delta^{2}M^{4}}{3}}}x}},$where Δ is a quantizer step size, M is an index assignment parameter and$\sigma_{X}^{2} = {\underset{- \infty}{\int\limits^{\infty}}{x^{2}{f_{X}(x)}{\mathbb{d}x}}}$is the variance of variable X that is related to the probability densityfunction ƒ_(X)(·) of the first probability density.
 13. A decoder forreconstructing a source signal, which is encoded by a set of at leasttwo descriptions, the source signal having a first probability density,wherein the first probability density comprises a first statisticalmoment and a second statistical moment of the first probability density,the decoder comprising: an input for receiving a subset of the set ofdescriptions; a reconstructor for reconstructing a reconstructed signalat an operating bitrate of a set of operating bitrates upon the basis ofthe subset of descriptions, the reconstructed signal having a secondprobability density, wherein the second probability density comprises afirst statistical moment and a second statistical moment of the secondprobability density; and a transformer for manipulating thereconstructed signal in order to obtain a manipulated reconstructedsignal having a third probability density, wherein the third probabilitydensity comprises a first statistical moment and a second statisticalmoment of the third probability density, wherein the transformer isconfigured to manipulate the reconstructed signal such that at least thefirst statistical moment and the second statistical moment of the thirdprobability density are more similar to the first statistical moment andthe second statistical moment of the first probability density than thefirst statistical moment and the second statistical moment of the secondprobability density are, and wherein the transformer is furthermoreconfigured to manipulate the reconstructed signal such that,irrespective of the operating bitrate, a predetermined minimumsimilarity between the first statistical moment of the third probabilitydensity and the first statistical moment of the first probabilitydensity and between the second statistical moment of the thirdprobability density and the second statistical moment of the firstprobability density is maintained.
 14. The decoder of claim 13, whereinthe transformer is configured to manipulate the first statistical momentand the second statistical moment of the second probability density inorder to preserve the first statistical moment and the secondstatistical moment of the first probability density within apredetermined moment range.
 15. The decoder of claim 13, wherein thereconstructor comprises a central reconstruction path, which isconfigured to reconstruct the reconstructed signal upon the basis ofindex information, the central reconstruction path comprising an indexerconfigured to determine the index information upon the basis of the setof descriptions.
 16. The decoder of claim 13, wherein the reconstructorcomprises a central reconstruction path, which is configured toreconstruct the reconstructed signal upon the basis of unique indexinformation, the central reconstruction path comprising an indexerconfigured to determine the unique index information upon the basis ofthe set of descriptions.
 17. The decoder of claim 13, wherein thereconstructor comprises at least one side reconstruction path, which isconfigured to reconstruct the reconstructed signal upon the basis ofmapping information, the at least one side reconstruction pathcomprising a mapper configured to determine the mapping information uponthe basis of the descriptions of the subset and of a composition ofdescriptions in the subset.
 18. The decoder of claim 13, wherein thetransformer is configured to perform the manipulating upon the basis ofa composition of descriptions in the subset of descriptions.
 19. Atleast one processor configured to reconstruct a source signal, which isencoded by a set of at least two descriptions, the source signal havinga first probability density, wherein the first probability densitycomprises a first statistical moment and a second statistical moment ofthe first probability density, by: receiving a subset of the set ofdescriptions; reconstructing a reconstructed signal at an operatingbitrate of a set of operating bitrates upon the basis of the subset ofdescriptions, the reconstructed signal having a second probabilitydensity, wherein the second probability density comprises a firststatistical moment and a second statistical moment of the secondprobability density; and manipulating the reconstructed signal in orderto obtain a manipulated reconstructed signal having a third probabilitydensity, wherein the third probability density comprises a firststatistical moment and a second statistical moment of the thirdprobability density, wherein the reconstructed signal is manipulatedsuch that at least the first statistical moment and the secondstatistical moment of the third probability density are more similar tothe first statistical moment and the second statistical moment of thefirst probability density than the first statistical moment and thesecond statistical moment of the second probability density are, andwherein the reconstructed signal is manipulated such that, irrespectiveof the operating bitrate, a predetermined minimum similarity between thefirst statistical moment of the third probability density and the firststatistical moment of the first probability density and between thesecond statistical moment of the third probability density and thesecond statistical moment of the first probability density ismaintained.
 20. The processor according to claim 19, wherein the firststatistical moment and the second statistical moment of the secondprobability density are manipulated to preserve the first statisticalmoment and the second statistical moment of the first probabilitydensity within a predetermined moment range.